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I have the following question: if I have an irregular symmetric polygon, how can I determinate the circumference with the least area that contains this polygon? I believe (in case that the polygon have only one axis of symmetry) that the center of that circumference belongs to the axis of symmetry of the polygon considered. Thanks for any suggestions, hints and more!

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You are right.

We can assume that our polygon $P$ is convex, since if $P\subset\Gamma $ for some circle $\Gamma$, then $\operatorname{Conv}(P)\subset\Gamma$, too. Assuming that $\Gamma_1$ is a minimal solution with its center outside the axis of symmetry, then the symmetric of $\Gamma_1$ with respect so such axis, $\Gamma_2$, is also a minimal solution, and $P\subset\Gamma_1\cap\Gamma_2$. However, the diameter of $\Gamma_1\cap\Gamma_2$ is stricly less than the diameter of $\Gamma_1$, contradicting the minimality, since the circle having the vertices of $\Gamma_1\cap\Gamma_2$ as endpoints of a diameter would be smaller than $\Gamma_1$ and contain $P$.

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